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Homology of the Torelli group and negative-dimensional vector spaces

OK, not really. You know and I know there’s no such thing as a negative-dimensional vector space.

And yet…

The Torelli group T_g is a subject of hot interest to mapping class groups people — it’s the kernel of the natural surjection from the mapping class group Γ_g to Sp_{2g}(Z). You can think of it as “the part of the mapping class group that arithmetic lattices can’t see,” or at least can’t see very well, and as such it is somewhat intimidating. We know very little about it, even in small genera. One thing we do know is that for g at least 3 the Torelli group is finitely generated; this is a theorem of Johnson, and a recent paper by Andy Putman provides a small generating set. So H_1(T_g,Q) is finite-dimensional. (From now on all cohomology groups will be silently assigned rational coefficients.)

But a charming argument of Akita shows that, in general, T_g has some infinite-dimensional homology groups. How do we know? Because if it didn’t, you would be able to compute the integer χ(T_g) from the formula

χ(T_g) = χ( Γ_g)/ χ(Sp_{2g}(Z)).

But both the numerator and denominator of the right-hand-side are known, and their quotient is not an integer once g is at least 7. Done!

At the Park City Mathematics Institute session I visited this summer, there was a lot of discussion of what these infinite-dimensional homology groups of Torelli might look like. We should remember that the outer action of Sp_2g(Z) on Torelli yields an action of Sp_2g(Z) on the homology of Torelli — so one should certainly think of these spaces as representations of Sp_2g(Z), not as naked vector spaces. In the few cases these groups have been described explicitly, they are induced from finite-dimensional representations of infinite-index subgroups H of Sp_2g(Z).

I just wanted to record the small observation that in cases like this, there’s a reasonably good way to assign a “dimension” to the homology group! Namely: suppose G is a discrete group and H a a subgroup, and suppose that both BG and BH are homotopic to finite complexes. (This is not quite true for G = Sp_2g(Z), but surely you’re willing to spot me a little finite level structure wherever I need it.) Let W be a finite-dimensional representation of H and let V be the induction of W up to G.

Now if H were finite-index in G you’d have

dim V = [G:H] dim W

or, what’s the same,

dim V = χ(BH)/χ(BG) dim W

But note that the latter formula makes sense even if H is infinite-index in G! And this allows you to assign a “dimension” to some infinite-dimensional homology groups.

For instance, consider T_2, which is not finitely generated. By a theorem of Mess, it’s a free group on a countable set of generators; these generators are naturally in bijection with cosets in Sp_4(Z) of a subgroup H containing SL_2(Z) x SL_2(Z) with index 2. Compute the Euler characteristics of H and Sp_4 and you find that the “dimension” of H_1(T_2) is -5.

And when you ask Akita’s argument about this case, you find that the purported Euler characteristic of T_2 is 6; a perfectly good integer, but not such a great Euler characteristic for a free group to have. Unless, of course, it’s a free group on -5 generators.

If you want to see this stuff written up a bit (but only a bit) more carefully, here’s a short .pdf version, which also includes a discussion of the hyperelliptic Torelli group in genus 3.

4 thoughts on “Homology of the Torelli group and negative-dimensional vector spaces”

Cool indeed!
Note that Lubotzky-Meiri (arXiv:1104.2450) and Malestein-Souto have shown that one can use actions of the Torelli group on the homology of double covers of the surface to investigate at least certain questions… (E.g., this is enough to detect pseudo-Anosov’s and show they are generic in a strong sense.)

Do you know the great paper “Negative sets have Euler characteristic and dimension” by Steve Schanuel? His model for negative sets is a certain category of polyhedra, the “negativity” being in their Euler characteristic. The dimension is the geometric dimension (as opposed to the algebraic dimension that you’re dealing with). It’s a really deft piece of work.

He argues – convincingly, I think – that Euler characteristic for geometric objects is closely analogous to cardinality of sets. The analogue for vector spaces is surely dimension, in which case negative-dimensional vector spaces occupy a similar conceptual position to negative (that is, negative-cardinality) sets. He writes:

“Euler’s analysis, which demonstrated that in counting suitably ‘finite’ spaces one can get well-defined negative integers, was a revolutionary advance in the idea of cardinal number — perhaps even more important than Cantor’s extension to infinite sets, if we judge by the number of areas in mathematics where the impact is pervasive.”

Here is a very small constraint suggested by the expressions in terms of zeta values:
chi(Sp(2g,Z))=zeta(-1)zeta(-3)…zeta(1-2g) and
chi(M_g)=zeta(1-2g)/(2-2g), hence
chi(I_g)=[(2-2g)zeta(-1)…zeta(3-2g)]^-1. The 2-2g probably cancels, so all that matters are zeta numerators, which often don’t cancel.

Inducing from small arithmetic groups to the symplectic group will yield similar reciprocal zeta values. So Akita does not rule out that possibility. But such induced modules have too many denominators: they always have the last value zeta(1-2g), which does not appear in chi(I_g). In particular, when g=6, chi(I_g) is an integer, but the induced modules we expect involve zeta(-11) and thus have denominators of 691. They have to add up in the right way to give an integer as the final answer. (One could compute the Euler characteristic modulo 1 as a function of homology not as vector spaces, but as vector spaces modulo finite dimensional vector spaces.)

Similarly, in your note you say “if H is any arithmetic sublattice of Sp_14(Z) which is not actually finite-index, the induced representation Ind_H^Sp_14 Q will also have virtual dimension with a denominator of 691.” I’m not sure whether that is supposed to be a lower or upper bound on the denominator, but if H=Sp_12 or Sp_12xSp_2 (and probably no other), the virtual dimension is an integer.